a general approach to multicomponent distillation column design
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Ingeniería QuímicaTRANSCRIPT
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A General Approach To Multicomponent Distillation Column Design
By 1Afolabi, Tinuade Jolaade,
Department of Chemical Engineering,
Ladoke Akintola University of Technology
P.M.B. 4000, Ogbomoso. Nigeria
Phone number 08036669764
And
Denloye, Adetokunbo
Department of Chemical Engineering,
University of Lagos, Akoka,
Nigeria.
1Corresponding author
2
Abstract
The application of the developed vapour-liquid equilibrium analytical correlations to
computer aided multicomponent distillation designs showed that it is not enough that a
property model (VLE) predicts well, the model must also have continuous first order
derivative to be useful in computer aided design. A general procedure therefore has been
developed that can be used for complex systems like petroleum compounds by modifying the
“θ-method” of convergence to handle situation in which discontinuity can be encountered.
Keywords: Multicomponent Distillation, Thermodynamic model, θ-method, Simulation
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1. Introduction
In chemical technology, separation of the constituents of a homogenous mixture is
one of the most prominent problems encountered and it can be solved either by the
introduction of a phase, immiscible or partially miscible, with one of the constituents or by
creation of a second phase either by heating (i.e. vaporization) or by condensation.
Multicomponent distillation, which is the most dominant separation process, utilizes the latter
method. It is the separation of a liquid mixture based on the differences in the volatilities of
the liquid constituents.
The simulation of chemical processes like multicomponent distillation requires that
thermo-physical property equations be solved together with the mass balance, energy balance
and other design equations. The relationship between these thermo-physical properties and
process variables are usually non-linear and in many cases include implicit functions. Since
these equations cannot be solved analytically, they are done numerically by trial and error
method which often leads to a convergence problem.
Traditionally, these thermo physical properties like vapour-liquid equilibrium (VLE)
and enthalpy are provided by specialized subroutines. Numerous subroutines are made
available to cover different ranges of temperature and pressure and different types of
mixtures [1]. A collection of these subroutines is known as thermodynamic and physical
properties data base (TPPD). The subroutines are written to provide “point values” of these
properties at specified conditions and incorporated in the process calculation. However, the
dependence of these properties on temperature, pressure and composition tends to be
neglected when only point values are used in the computational process.
Holland [2] developed certain equations for accelerating or inducing convergence in
the solution of multicomponent distillation problems. In practice, this procedure has been the
most successful of any adjunct to the basic Thiele-Geddes or Lewis-Matheson procedure for
solving multicomponent distillation problems [3]. This paper is aimed at modifying the
Holland’s θ- method of convergence to solve the combined rigorous thermo- physical
property equations with the process model equations, therefore making it a general approach.
2. State of the Art
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2.1 Model Equations
Three sets of broad equations are used to model a multicomponent distillation column,
namely, equilibrium relations, material balance and enthalpy balance. The three sets of
equations are:
Equilibrium Relationship
Njx
Njy
Njxky
c
iij
c
iji
jijiji
2,11
2,11
2,1
1
1 (1)
Material Balance
BiDii
Bijijjj
DiiffFiFfif
Dijijjj
BxDxFXNjBxxLyV
DxxLyVyVfjDxxLyV
2,2,1
2,2,1
11
,11
11
(2)
Enthalpy Balance
BDF
Bjjjj
cDiffFFiffi
cDjjjj
BhDHFHNjBhhLHV
QDxhLHVHVfjQDHhLHV
2,2,1
2,2,1
11
11
11
(3)
2.2 Solution Procedures
In the solution of a set of nonlinear equations by iterative techniques, convergence or
divergence of a given calculational procedure depends not only on the initial choice of the
independent variables but also on the precise ordering and arrangement of each equation of
the set. Several authors [4, 5, 6, 7, 8] have developed many calculational procedures for
multicomponent distillation. For example, Amundson and Pontinen [9] developed a matrix
method; Holland [2] developed a -method; Wang and Henske [10] developed a tridiagonal
matrix method and Naphtali and Sandholm [11] developed a linearization method.
These methods have both advantages and disadvantages. For example, Naphtali and
Sandholm’s [11] method is excellent in the ability to converge, but it requires a large
computer memory and the number of calculations per trial is very large. It is suited, therefore
for large central system memory. Wang and Henske’s [10] method does not require a large
memory. The equations are simple and the number of calculations per trial is small, but its
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ability to converge calculation is not strong. It is suited, accordingly, for interactive terminal
users or microcomputer users. Holland’s -method has been applied in various versions with
considerable success. However, point values of the thermodynamic properties used in the
computational process are continuous functions even at higher derivatives to ensure
convergence.
The calculational procedure for the -method is summarized as follows:
1. Assume a set of temperatures (Tj) and a set of vapour rates (Vj). The set of liquid rates
corresponding to the set of assumed vapour rates are found by use of the total material
balance equation
2. Calculate the vapour-liquid equilibrium constant of each component.
3. On the basis of the temperatures and flow rates assumed in step 1, solve the material
balance equations for the composition of each component at each stage.
4. Determine each stage temperature.
5. Compute the phase (Total) flow rate using the energy balance equations
6. Repeat steps (2) – (5) until Tnjnj ETT 1
for each stage where ET is a prescribed
tolerance.
This work modified this -method to accommodate the rigorous thermodynamic models.
2.3 Thermodynamic Model
Equations of state have enjoyed the widest application in distillation
calculations. Notable are Benedict-Webb-Rubin (BWR EOS), Redlich-Kwong (RK EOS),
Soave-Redlich-Kwong (SRK EOS) and Peng-Robinson (PR EOS) equations of state [12]. All
the equations needed to compute the VLE values and enthalpies by use of these models are in
the literature [13, 14]. VLE are expressed in terms of fugacity as
ffK v
i
li
i (4)
where the mixture fugacity coefficient fl is for the liquid and fv for the vapour.
Unlike the original BWR EOS, which is recommended for computing the properties of
both vapour and liquid phases, the RK EOS is recommended for computing properties of
only the vapour phase. The SRK EOS and the PR EOS are recommended for computing the
properties of both phases for certain pure components and mixtures [12, 13, 15]. These
thermodynamic models have the following advantages over the other models
• They predict VLE values for a wider range of pressure with good precision.
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• The only parameters needed for predictions are the reduced temperature ,Tr; reduced
pressure, Pr; and the accentric factor,
• These parameters can be generated for any petroleum compound including crude oil
that is characterized by normal boiling point (NBP) and specific gravity (SG) [14, 16].
• They can also be used to calculate the departure enthalpy (residual enthalpy) needed
for vapour and liquid enthalpies.
The American Petroleum Institute’s Technical Book, “Petroleum Refining”, adopted the SRK
EOS procedure for the VLE calculation [17]; therefore it was utilized in this work.
3. The Modified Solution Procedure
The stage temperature determination is the least satisfactory part of any method of
solution and this is the case with -method of convergence [18]. In the conventional bubble
point method of iterative distillation calculation, the temperature profile in the column is
estimated on the basis of the liquid composition at each stage by bubble point temperature
calculation or the vapour composition by dew point calculation. Wang and Henske [10]) used
Muller’s method to obtain bubble point temperature, while most of the other investigators
used the Newton- Raphson iterative method. Holland [2] suggested that temperature for each
stage can be approximated by the Kb method.
The procedure was modified at this stage temperature determination step. A more recent
concept of process simulation that combines the rigorous thermo physical property equation
with the process model equation was utilized, with a little modification to reduce the several
problems involved in the approach. The number of equations is increased and this enlarged
set of equation poses a difficult problem to solve due to the highly non-linear nature of most
thermo-physical properties. Also, the analytical evaluation of the numerous partial derivatives
can be cumbersome since there will be a need for redevelopment each time a new or different
property relation is used.
Therefore, the solution method had to be modified to avoid derivatives of K since the
simulation will not converge if the first derivative of the property models is not continuous.
In order to achieve this, a subroutine based on thermodynamic equilibrium was used to
calculate the bubble point temperature as shown in Fig. 1.
This modified procedure though iterative in nature, is sequential and does not involve
partial derivatives of the thermo-physical model. Normally in simulation and design
problems, one or more of the state variables (e. g. temperature and/or vapour composition) is
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unknown and iterative procedure generates temporary values for these variables. Thus
property models must be continuous and well behaved at conditions that are not “real” to
achieve convergence [19]. Example of property models that can cause discontinuity in
“unreal” conditions are the equation of state model. For instance, if non-condensable
compounds are present in a mixture, the equation of state may fail to give the required liquid
root during iterative solution of a flash calculation.
It was pertinent to use a procedure that does not involve derivatives because Gani and O’
Connel [20] have pointed out that property models are often the cause of non-linear process
models and therefore the cause of difficulty in achieving convergence. It was categorically
stated that if the first derivative of the property models are not continuous the simulation will
not converge. Consequently, in a process design and simulation, property models are required
to have continuous first-order derivatives.
Figure 1. Flow diagram of the algorithm using SRK EOS for stage temperature calculation
Input data
Guess value of T for first iteration
Assume
Calculate fli (T, P, xi )
(i=1,2………n)
Calculate vi (T, P, yi ) (i=1 2………n)
Is SUMY constant?
Print Ti and yi
1
Read P and xi
Pfyi
li
i
iySUMY
First iteration
Is SUMY=1
Readjust T
NO
YES
NO
YES
NO
YES
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4. Simulation Examples, Results and Discussion
Two multicomponent distillation design problems of different complexity described
by Holland (1981) were solved using the modified procedure. Case Study 1 is a conventional
multicomponent distillation problem with one inlet stream and two outlet streams while Case
Study 2 is a complex multicomponent distillation problem with a side stream.
The modified theta method of convergence was used to solve the resulting non-linear
sets of equation and compared with Holland’s thereby checking its effect on the feed flash
temperature, vapour and liquid distribution along the column and the rate of convergence.
4.1 Case Study 1
The calculated flash temperatures of the feed using Holland’s is 624.44 oR and for this
work is 641.24 oR respectively. The results are close despite the fact that this work did not use
the “point values” of these properties at specified conditions. Figures 2 - 4 show the
temperature, liquid and vapour flow rates profiles along the stages in a conventional
distillation column.
Figure 2 Temperature profile for Case Study 1
0
100
200
300
400
500
600
700
800
900
0 2 4 6 8 10 12 14
Stages
Tem
pera
ture
(oR
)
Holland's Method
This Work
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Figure 3 Liquid flowrate profile for Case Study 1
0
50
100
150
200
250
0 2 4 6 8 10 12 14
Stages
Liqu
id fl
owra
te(m
ol/h
)
Holland's Method
This Work
In all the Figures, the shapes of the temperature, liquid flowrate and vapour flowrate
profiles are the same and the correlation coefficients (R2) are 0.9989, 0.9941 and 0.9993
respectively. The two methods converged at different rates; the modified procedure (15 trials)
converged faster than Holland’s (25 trials).
Figure 4 Vapour flowrate profile for Case Study 1
0
20
40
60
80
100
120
140
160
0 2 4 6 8 10 12 14
Stages
Vap
our f
low
rate
s(m
oll/h
)
Holland's MethodThis Work
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4.2 Case Study 2
Figures 5 – 7 show the temperature, liquid flowrate and vapour flowrate profiles in a
complex distillation column.
Figure 5 Temperature profile for Case Study 2
0
50
100
150
200
250
300
350
400
450
500
0 2 4 6 8 10 12 14
Stages
Tem
pera
ture
(oR)
Holland's MethodThis Work
In all the Figures, the shapes of the profiles are the same and the correlation coefficients
(R2) are 0.9945, 0.9970 and 0.9811 respectively. The calculated flash temperatures of the
feed using Holland’s is 624.44 oR and for this work is 641.24 oR respectively. These show
that the feed temperature is not affected by the complexity of the column, since the calculated
values are the same for the two cases studied. This is expected because the feed composition
and its conditions are the same for the two cases. It was also observed that as in Case 1, the
two methods converged at different rates even though the convergence is dependent on two
variables, 1and 2 instead of only one in Case 1.
The trend in the results obtained in the two cases is similar. However, the deviation
from the expected is less for the temperature profile and more for the vapour and liquid
flowrate profiles in the complex column than the conventional column. This may be caused
by the complexity of Case 2.
5. Conclusion
The Theta method of convergence has been modified to handle situation in which
discontinuity can be encountered and a general procedure has been developed that can be
used for complex systems like petroleum compounds since the only parameters needed for
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predictions are the reduced temperature ,Tr; reduced pressure, Pr; and the accentric factor, .
These parameters can be generated for any petroleum compound including crude oil that is
characterised by normal boiling point (NBP) and specific gravity (SG).
NOTATIONS
c Total number of component
D Total molar flow rates of the distillate, mol/h
F Total molar flow rates of the feed
HFi, hFi Enthalpy of pure component i evaluated at the temperature TF and pressure P
of the flash, Btu/mol
Hi
c
ijiji yH
1, for an ideal solution; at the temperature Tj,
hi
c
ijiji xh
1,for an ideal solution; at the temperature Tj,
Kji Equilibrium vapourization constant;
Lj Total molar flow rates of liquid leaving any stage of j, mol/h
Lji Molar flow rates at which component i in the liquid phase leaving the jth plate,
mol/h
N Total number of stages
Qc Condenser duty, Btu/h
QB Reboiler duty, Btu/h
Tj Temperature of any stage, oR
Vj Total molar flow rates of vapor leaving any stage of j, mol/h
Vji Molar flow rates at which component i in the vapour leaveing plate j, mol/h
xi Mole fraction of component i in liquid phase for any plate
yi Mole fraction of component i in vapour phase for any plate
Subscript
f Feed plate
F Variable associated with partially vapourized feed
i Component number i=1,2,…
j Stage number
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